Elements in the component Max\((S)\)  have a number of related geometric interpretations. Schoen [1]} has shown these representations correspond to equivariant minimal Lagrangians in \(\mathbb {H}^{2} \times \mathbb {H}^{2}\) . At the same time, the group \(G=\mathrm {PSL}(2,\mathbb {R})\times \mathrm {PSL}(2,\mathbb {R})\)  is the isometry group of AdS\(^{3}\) , and Mess [2]} has shown the holonomy representations of GHMC-AdS\(^{3}\)  manifolds are precisely the ones in Max\((S)\) . Krasnov-Schlenker [3]} have shown to each GHMC-AdS\(^{3}\)  manifold, there is a unique equivariant space-like maximal surface, whose image under the Gauss map is the aforementioned minimal Lagrangian.
